First, I don't think it's easily possible. All launched programs show on the launcher by default, some like pidgin, will disappear if you close the window but still remain on the panel(if whitelisted) but if they are not designed to do that, getting it to happen is likely to involve recoding part of Steam. Second, even if yo could get it to disappear, it would come back as soon as you opened a game(if you notice when a game is running there are 2 arrows beside the icon) so that nothing would be gained by doing all the work to get it to go away.
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TrailRiderJul 11 '13 at 4:29

Yeah I want Steam to behave like Pidgin or Transmission which goes to panel, not to launcher,when the window closed, whats the point of close button anyway if its functionality is same as minimize. But looks like it cant be done. Thanks for answering.
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NurJul 11 '13 at 9:54

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The closest you can come is to launch your games from the dash. It will launch Steam only in the panel. You can then launch other games by clicking on the icon and then selecting the game form the pop up menu. However if you open the Steam Client you cannot close it again and have it disappear from the launcher. It doesn't matter how you do it as far as memory is concerned unless you are very lean on memory, there is only a 30-50MB difference so it will not help much. 140MB compared to 180MB when I tested it on my computer.
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TrailRiderJul 11 '13 at 22:37

I posted the last comment as a comment instead of an answer because it doesn't really answer your question, It is as close to an answer as you can get I think. I.E. your question was "HOW" to remove the icon as opposed to "IF" you can. so my answer of "NO" does not really answer it. Wish I could have been of more help.
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TrailRiderJul 11 '13 at 22:40

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Go complain to valve. Seriously, write some sort of bug report or something, because this is actually a problem. The windows version of steam does not suffer from this issue - closing steam there does indeed remove its 'open window' tab thing from the main bar, but leaves the icon in the taskbar. Why should the linux version need users to perform some big workaround to fix this issue?
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AJMansfieldAug 17 '13 at 18:58

I just got this to work by reading how some Windows users got it to work.
You just need to add "-sllent" to the startup command.
I did this by using Startup Applications.
Scroll down till you see Steam, select it and then click Edit. Change the command to look like this:

/usr/bin/steam %U -silent

I think this also removes it from your Unity panel which is OK by me. Seems redundant to have it in both the tray and the panel anyways.